Stability of reactive shocks in the generalized Noh problem

D'yakov-Kontorovich (DK) instability is the name of the instability associated with steady shock waves. Despite the extensive literature accumulated since the pioneering works in the 1950s, the stability of steady shocks is still an open question when realistic boundary conditions are considered. There is just one explicit dispersion relationship known so far that applies only to isolated shocks, for which the DK instability reduces to a non-decaying oscillating regime accompanied by a spontaneous acoustic emission (SAE). However, the consideration of a supporting mechanism modifies the shock dynamics in this unstable range. We derive an explicit equation that determines the stability limits and the growth rate of perturbations for steady expanding shocks provided by the generalized Noh problem, i.e., non-isolated conditions. The dispersion equation is presented in terms of fundamental parameters describing the shock and geometry parameters. Within the SAE conditions used by Bates and Montgomery in a van der Waals gas, we find that cylindrical and spherical expanding shocks become literally unstable for sufficiently high mode numbers. Counterintuitively, it is found that the effect of exothermicity or endothermicity across the shock is stabilizing or destabilizing, respectively.